∫ x^6(5x-6) wrt x eching

$$\int x ^ { 6 } ( 5 x - 6 ) d x$$

Baholash

$$\frac{5x^{8}}{8}-\frac{6x^{7}}{7}+С$$

Yechim qadamlari

x^{6} ga 5x-6 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\int 5x^{7}-6x^{6}\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int 5x^{7}\mathrm{d}x+\int -6x^{6}\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$5\int x^{7}\mathrm{d}x-6\int x^{6}\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{7}\mathrm{d}x integralni \frac{x^{8}}{8} bilan almashtiring. 5 ni \frac{x^{8}}{8} marotabaga ko'paytirish.

$$\frac{5x^{8}}{8}-6\int x^{6}\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{6}\mathrm{d}x integralni \frac{x^{7}}{7} bilan almashtiring. -6 ni \frac{x^{7}}{7} marotabaga ko'paytirish.

$$\frac{5x^{8}}{8}-\frac{6x^{7}}{7}$$

Агар F\left(x\right)f\left(x\right) ning dastlabki holati boʻlsa, u holatda f\left(x\right) ning barcha dastlabki holatlari toʻplami F\left(x\right)+C tarafidan belgilanadi. Shu sababli natijaga C\in \mathrm{R} integrallash konstantasini qoʻshing.

$$\frac{5x^{8}}{8}-\frac{6x^{7}}{7}+С$$

x ga nisbatan hosilani topish

$$\left(5x-6\right)x^{6}$$